= Matsubara derivation of the BCS saddle
{c}
{title2=$\beta^{-1}\sum_n(\omega_n^2+E^2)^{-1}=\tanh(\beta E/2)/(2E)$}
Differentiate the <subtracted grand potential of the reduced BCS model> with respect to the squared pairing amplitude. The displayed fermionic <Matsubara sum> converts the derivative of the trace logarithm into $\sum_k\tanh(\beta E_k/2)/(2E_k)$. Setting it equal to $V/\lambda$ gives the uniform nonzero-gap saddle. The limit E→0 is finite at positive temperature, $\beta/4$. The sum relies on antiperiodic temporal boundary conditions from the <fermionic coherent-state trace>.
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